Explore BrainMass

Explore BrainMass

    Area under a function

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    Consider a continuous, positive function f: R --> R. The graph of f(x) is the set of points in the (x,y) Cartesian plane such that y = f(x).

    See Attached file for Figure and questions.

    © BrainMass Inc. brainmass.com October 7, 2022, 5:06 pm ad1c9bdddf
    https://brainmass.com/math/graphs-and-functions/area-under-function-218448

    Attachments

    SOLUTION This solution is FREE courtesy of BrainMass!

    Please see the attachment.

    We kow that the area can be represented as a definite integral as follows.
    .
    (a) Since , , then for any , we have . Then we have
    and
    Thus
    (b) Set . From (a), we get . Since is a continuous function from the graph, then can reach maximum and minimum in the closed interval . Then we can find some , such that and . By the Intermediate Value Theorem, we can find some between and , such that . Then we have , where is between and , hence is in .
    (c) We note that represent the area between and , according to (b), we can find some , such that . Therefore, we get .
    (d) From part (c), we have , then we get

    Let , then and the right side is the definition of . Then we have

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com October 7, 2022, 5:06 pm ad1c9bdddf>
    https://brainmass.com/math/graphs-and-functions/area-under-function-218448

    Attachments

    ADVERTISEMENT